Increase spatial sampling for wave front mid-spatial frequency error recovery

ABSTRACT

A differential curvature sensing device for measuring a wavefront curvature by employing increased spatial sampling for wavefront testing with mid-frequency error recovery. The device includes a sampling sensor having an output beam, an optical element to split said output beam, a lenslet array in the path of each beam to generate corresponding sampling grids, a shearing element for shifting the grid points in horizontal and vertical directions to produce plural sampling grids having plural grid points for use generating a spatial sampling grid having a density for mid-spatial frequency recovery. The displacement of the shifting less than a pitch size of the lenslet array, and a measuring device measuring plural slopes of plural wavefronts at each grid point to obtain a wavefront normal curvature and corresponding twist curvature terms to determine a principal curvature and directions. The sensor is a Shack-Hartman sensor, shearing interferometer sensor and other discrete-point sampling sensors.

This application is a continuation-in-part of U.S. patent Ser. No. 11/414,586 filed on Apr. 28, 2006 and claims the benefit of priority to U.S. Provisional Patent Application No. 60/692,884 filed on Jun. 22, 2005.

FIELD OF THE INVENTION

This invention relates to sensors and, in particular, to methods, systems, apparatus and devices for a differential Shack-Hartmann curvature sensor to measure the differentials of wave front slope, i.e. wave front curvatures, to determine the wave front shape and for wave front testing with mid-spatial frequency error using increased spatial sampling for wave front mid-spatial frequency error recovery.

BACKGROUND AND PRIOR ART

The Shack-Hartmann sensor is one of the most popular wavefront sensors presently available. The sensor measures the slope data of the wavefront by comparing the coordinates of Hartmann grid points from the measurement beam with coordinates from the reference beam. FIG. 1 is a schematic layout of a prior art Shack-Hartmann sensor. Hartmann invented the Hartmann test in 1900, and Roland Shack improved this technique by introducing a lenslet array in 1971. The reference beam was generated by a pin-hole light source and injected into this system as ideal wavefront to calibrate the systematic error. In FIG. 1, reference beam 105 is represented by a dashed line, and the continuous line represents the measurement beam 120. The wavefront is first collimated by the collimation lens then passes through a lenslet array 110. The Hartmann screen divides the wavefront into many sub-apertures, then the micro-lens array 110 focus the wavefront of each sub-aperture into a group of Hartmann grid points. Comparing the coordinate differences between the measured wavefront and the reference wavefront, the wavefront slopes can be calculated according to the following formula:

$\begin{matrix} {\begin{matrix} {{\frac{\partial W}{\partial x}❘_{i}} = \frac{x_{i}^{mea} - x_{i}^{ref}}{f}} \\ {{\frac{\partial W}{\partial y}❘_{i}} = \frac{y_{i}^{mea} - y_{i}^{ref}}{f}} \end{matrix},} & (1) \end{matrix}$ where (x_(i) ^(ref), y_(i) ^(ref)) (i=1, 2, . . . , m, m=t×t is the total number of grid points) is the Hartmann grid coordinates of the reference beam, (x_(i) ^(mea), y_(i) ^(mea)) is the Hartmann grid coordinates of the measurement beam, and f is the focal length of the lenslet array.

Compared to the Hartmann test, the Shack-Hartmann wavefront sensor provides improved photon efficiency because the position of the focal spot is proportional to the average wavefront slope over each sub-aperture and the position is independent of higher-order aberrations and intensity profile variations. The Shack-Hartmann sensor is a parallel wavefront sensor operating in real time. It has application in active/adaptive optics, optical testing, opthalmology, telescope image analysis and atmosphere and random media characterizations.

However, the usage of the Shack-Hartmann sensor is restrictive because it is sensitive to vibration, tilt and whole body movement. Therefore, a need exists for a sensor that provides the benefits of the Shack-Hartmann sensor without the limitations.

SUMMARY OF THE INVENTION

A primary objective of the invention is to provide a new method, system, apparatus and device for measuring differentials of the wave front slopes to determine the wave front curvature without the use of an external reference or reference light source after calibration.

A secondary objective of the invention is to provide a new method, system, apparatus and device measuring differentials of the wavefront slopes to determine the wavefront curvature independent of vibration, tilt or whole body movement so it is useful for measurements on a moving stage.

A third objective of the invention is to provide a new method, system, apparatus and device for a differential curvature sensor that is scale tunable by varying differential values.

A fourth objective of the invention is to provide a new method, system, apparatus and device for a differential curvature sensor that can be applied in optical testing with vibration.

A fifth objective of the invention is to provide a new method, system, apparatus and device for a differential curvature sensor without moving parts for increased reliability.

A sixth objective of the invention is to provide a new method, system, apparatus and device for a differential curvature sensor for use in optical testing, active and adaptive optics, shape-extraction in bio-optics and opthalmology, such as corneal measurement.

A seventh objective of the invention is to provide a new method, system, apparatus and device for a differential curvature sensor for measuring the normal curvatures and the twist curvature terms.

An eighth objective of the invention is to provide a new method, system, apparatus and device for a differential curvature sensor for measuring the wave front principal curvatures and directions.

A ninth objective of the invention is to provide a new method, system and apparatus for increased spatial sampling for mid-spatial frequency error recovery.

A first preferred embodiment of the invention provides a differential curvature sensing device for measuring a wavefront curvature using increased spatial sampling for wavefront testing with mid-frequency error recovery. The device includes a sampling sensor having an output beam, an optical element to split said output beam into a first, second and third beam, each of said first, second and third beams traveling in one of an x, y and z direction, respectively, and a first, second and third lenslet array in the path of the first, second and third beam, respectively, to generate a corresponding first, second and third sampling grid each having plural grid points. A shearing element for shifting said plural grid points a distance in a horizontal and a vertical direction to produce plural sampling grids each having plural grid points for generating a spatial sampling grid having a density for mid-spatial frequency recovery. The displacement of the shifting is less than a pitch size of the lenslet array, and a measuring device for measuring plural slopes of plural wave fronts at each grid point of each spatial sampling grid to obtain a wave front normal curvature and corresponding twist curvature terms for determining the principal curvatures and directions, wherein said plural slopes at each of the plural grid points are measured approximately simultaneously before and after differentially shifting the sampling grid. The sampling sensor is selected from a group including a Shack-Hartman sensor, a curvature sensor, a shearing interferometer sensor and a 4-dimensional technique.

A second preferred embodiment provides a system for measuring differentials of wave front slopes to determine a wavefront curvature without the use of a reference. The system includes a light source for generating an input light beam, a beam splitter splitting the input light beam to produce at least three output beams in different directions, at least three lenslet arrays, one in a path of each of the at least three output beams for generating a corresponding at least three sampling grids, at least one optical element for shearing two of the at least three output beams in a horizontal direction and a vertical direction.

A third preferred embodiment provides a method for measuring a normal curvature of a wavefront with mid-spatial frequency error detection with a differential curvature sensor including the steps of providing a sensing system having a first, second and third output beam in different directions for generating an original sampling grids having plural grid points, measuring differentials of the plural wave front slopes at plural grid points of the original sampling grid, moving the sampling grid in a horizontal direction with a displacement less than a pitch size of the lenslet compared to said original sampling grid, measuring differentials of the plural wave front slopes at plural grid points of the horizontally displaced grid to produce a second sampling grid, moving the original grid in a vertical direction with a displacement less than a pitch size of the lenslet compared to said original sampling grid to produce a third sampling grid, measuring differentials of the plural wave front slopes at plural grid points of the third sampling grid, repeating the horizontal and vertical moving and measuring steps or more times for increased spatial sampling, and determining said normal curvature of said wavefront from said measured differentials of said plural wave front slopes for wavefront testing with mid-frequency error recovery.

Further objects and advantages of this invention will be apparent from the following detailed description of preferred embodiments, which are illustrated schematically in the accompanying drawings.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 shows a prior art Hartmann screen and lenslet array and their coordinate differences.

FIG. 2 shows an example of a prior art Hartmann grid.

FIG. 3 shows a Shack-Hartmann grid with displacements in the x and the y directions.

FIG. 4 a shows a direct implementation layout of the differential Shack-Hartmann curvature sensor according to an embodiment of the present invention.

FIG. 4 b shows another direct implementation layout of the differential Shack-Hartmann curvature sensor according to another embodiment of the present invention.

FIG. 4 c shows yet another direct implementation of the differential Shack-Hartmann curvature sensor according to another embodiment of the present invention.

FIG. 5 shows a calibration block diagram according to the present invention.

FIG. 6 shows an experimental system for the differential Shack-Hartmann curvature sensor according to the first embodiment of the present invention.

FIG. 7 a shows the image space of Roddier's curvature sensing technique of the prior art.

FIG. 7 b shows the object space of Roddier's curvature sensing corresponding to the image space of FIG. 7 a.

FIG. 8 shows a sampling grid from a Differential Shack Hartmann sensor at (0, 0).

FIG. 9 shows the sampling grid of FIG. 8 with the grid shifted to by 0.6 along the x-axis to (0.6, 0).

FIG. 10 shows the sampling grid of FIG. 9 with the grid shifted to by 1.2 along the x-axis to (1.2, 0).

FIG. 11 shows the sampling grid of FIG. 10 with the grid shifted by −0.6 along the y-axis to (0, −0.6).

FIG. 12 shows the sampling grid of FIG. 11 with the grid shifted by 0.6 along the x-axis to (0.6, −0.6).

FIG. 13 shows the sampling grid of FIG. 12 with the grid shifted by 1.2 along the x-axis to (1.2, −0.6).

FIG. 14 shows the sampling grid of FIG. 13 with the grid shifted by −0.6 along the y-axis to (0, −1.2).

FIG. 15 shows the sampling grid of FIG. 14 with the grid shifted by 0.6 along the x-axis to (0.6, −1.2).

FIG. 16 shows the sampling grid of FIG. 15 with the grid shifted by 0.6 along the x-axis to (1.2, −1.2).

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Before explaining the disclosed embodiments of the present invention in details it is to be understood that the invention is not limited in its application to the details of the particular arrangements shown since the invention is capable of other embodiments. Also, the terminology used herein is for the purpose of description and not of limitation.

The following is a list of the reference numbers used in the drawings and the detailed specification to identify components:

100 Shack-Hartmann sensor 105 reference beam 110 Hartmann screen and lenslet array 120 measurement beam 200 Hartmann grid 310 Hartmann grid 320 horizontal displacement (S_(x)) 330 vertical displacement (S_(y)) 400 differential Shack-Hartmann curvature sensor 402 first lenslet array 404 second lenslet array 406 third lenslet array 410 beam splitters 412 first camera 414 second camera 416 third camera 417 lenslet shearing device (x) 418 lenslet shearing device (y) 610 first light source 620 second light source

The method, system, apparatus and device of the present invention provides a Shack-Hartmann curvature sensor that shears the Shack-Hartmann grid, horizontally and vertically, as shown in FIG. 3 to measure the wavefront curvatures at each Hartmann grid point. The curvature is the derivative of slopes. The differential Shack-Hartmann curvature sensor of the present invention is based on the prior art Shack-Hartmann sensor and the Hartmann grid shown in FIGS. 1 and 2, respectively. By measuring the differentials of the wavefront slopes at each Hartmann grid point, the wavefront local curvatures (normal curvatures and the twist term curvature) are determined, from which the principal curvatures and directions of the wave front are computed.

A. Normal Curvature Measurements

Differential displacements are made in the x and y directions to obtain slope differentials of the wavefront, the wavefront curvatures. The curvature sensor of the present invention is implemented as a Shack-Hartmann sensor with three output channels for achieving the Hartmann grid shearing in two perpendicular directions. The slope is measured in the three channels as well as the curvature of the wavefront at each Hartmann grid point.

The Hartmann grid is shifted a lateral differential distance in the x and in the y direction, which are suggested in this invention to adopt 1/10 to ½ of the pitch size of the Hartmann grid according to each specific application. As shown in FIG. 3, for example, the slopes at each point are measured before and after the differential shift to obtain the slope differentials, the curvature in the x direction and in the y direction can be obtained by applying

$\begin{matrix} {\begin{matrix} {\left. {{{{{c_{x}(i)} = \frac{\partial^{2}W}{\partial x^{2}}}}_{i} = {{\frac{1}{s_{x}}\left( \frac{\partial W}{\partial x} \right._{i^{\prime}}} - \frac{\partial W}{\partial x}}}}_{i} \right) = {{\frac{1}{f}\left( \frac{x_{i}^{{\prime\;}^{mea}} - x_{i}^{mea}}{s_{x}} \right)} - {c_{0,x}(i)}}} \\ {\left. {{{{{c_{y}(i)} = \frac{\partial^{2}W}{\partial y^{2}}}}_{i} = {{\frac{1}{s_{y}}\left( \frac{\partial W}{\partial y} \right._{i^{''}}} - \frac{\partial W}{\partial y}}}}_{i} \right) = {{\frac{1}{f}\left( \frac{y_{i}^{{''\;}^{mea}} - y_{i}^{mea}}{s_{y}} \right)} - {c_{0,y}(i)}}} \end{matrix},} & (2) \end{matrix}$ where c_(0,x)(i) and c_(0,y)(i) are obtained by

$\begin{matrix} {\begin{matrix} {{c_{0,x}(i)} = {\frac{1}{f}\left( \frac{x_{i}^{{\prime\;}^{Ref}} - x_{i}^{Ref}}{s_{x}} \right)}} \\ {{c_{0,y}(i)} = {\frac{1}{f}\left( \frac{y_{i}^{{''\;}^{Ref}} - y_{i}^{Ref}}{s_{y}} \right)}} \end{matrix},} & (3) \end{matrix}$ and in this paper S_(x) and S_(y) are the differential shifts in the x- and y-directions, respectively. Theoretically c_(0,x)(i) and c_(0,y)(i) are “1/f”, because the lenslet array is moved a lateral distance S_(x) in x-direction, for example, the reference Shack-Hartmann grid moves exactly the same distance s_(x) accordingly, therefore, x_(i)′^(Ref)−x_(i) ^(Ref)=s_(x) and c_(0,x)(i)=1/f; similarly, c_(0,y)(i)=1/f. However, in practice c_(0,x)(i) and c_(0,y)(i) are not “1/f”, because the coordinates are measured by different CCD cameras, so they belong to different coordinate system and therefore c_(0,x)(i) and c_(0,y)(i) are obtained by calibration.

The slope differentials in the cross directions, referred to as the corresponding twist curvatures, are given by

$\begin{matrix} {\begin{matrix} {\left. {{{{{c_{xy}(i)} = \frac{\partial^{2}W}{{\partial x}{\partial y}}}}_{i} = {{\frac{1}{s_{y}}\left( \frac{\partial W}{\partial x} \right._{i^{''}}} - \frac{\partial W}{\partial x}}}}_{i} \right) = {{\frac{1}{f}\left( \frac{x_{i}^{{''\;}^{mea}} - x_{i}^{mea}}{s_{y}} \right)} - {c_{0,{xy}}(i)}}} \\ {\left. {{{{{c_{yx}(i)} = \frac{\partial^{2}W}{{\partial y}{\partial x}}}}_{i} = {{\frac{1}{s_{x}}\left( \frac{\partial W}{\partial y} \right._{i^{\prime}}} - \frac{\partial W}{\partial y}}}}_{i} \right) = {{\frac{1}{f}\left( \frac{y_{i}^{{\prime\;}^{mea}} - y_{i}^{mea}}{s_{x}} \right)} - {c_{0,{yx}}(i)}}} \end{matrix},} & (4) \end{matrix}$ where c_(0,yx)(i) and c_(0,xy)(i) are constants given by

$\begin{matrix} {\begin{matrix} {{c_{0,{yx}}(i)} = {\frac{1}{f}\left( \frac{y_{i}^{{\prime\;}^{Ref}} - y_{i}^{Ref}}{s_{x}} \right)}} \\ {{c_{0,{xy}}(i)} = {\frac{1}{f}\left( \frac{x_{i}^{{''\;}^{Ref}} - x_{i}^{Ref}}{s_{y}} \right)}} \end{matrix},} & (5) \end{matrix}$ In Equations (2) and (4), (x_(i) ^(mea), y_(i) ^(mea)), (x_(i)′^(mea), y_(i)′^(mea)) and (x_(i)″^(mea), y_(i)″^(mea))(i=1, 2, . . . m) are the coordinates of the measured original Hartmann grid points, the measured sheared Hartmann grid points in the x-direction and in y-direction, respectively. When a differential shift S_(x) is made in x-direction, the y-coordinates remains unchanged, so c_(0,yx)(i)=0; similarly, c_(0,xy)(i)=0. However, in practice, these two constants may not be zero as the coordinates are measured by three CCD cameras, and they are usually obtained by calibration.

If the wavefront travels in the z-direction, and the beam is split into three parts traveling in three different directions, one beam travels in the z-direction, another beam travels in the x-direction, and the third beam travels in the y-direction. For example, three lenslet arrays 402, 404 and 406, are placed in the three beams such that 402 is in the z-direction beam, 404 is in the x-direction beam, and 406 is in the y-direction beam as shown in FIG. 4 a. Lenslet 404 is conjugated to 402, and 406 is conjugated to 402, separately. Using a micro-screw or a similar shearing device, lenslet array 404 is moved a differential distance in the x-direction, and using a second micro-screw or a similar shearing device, lenslet array 406 is moved a differential distance value in the y direction. Thus, the lenslet 404 and lenslet 402 are shearing each other in the x-direction, and lenslet 406 and lenslet 402 are shearing each other in the y-direction. By measuring the coordinates of Hartmann grid points generated by the three beams, and applying Equations (2) and (4), the wavefront curvatures in the x- and y-directions and the twist terms are obtained. In an example, the Hartmann grid coordinates are measured by placing a charge-coupled device (CCD) camera or similar device in the path of each of the three beams for recording the Hartmann grid points.

Examples of alternative direct implementation layouts are shown in FIGS. 4 b and 4 c. As shown in FIG. 4 b, optical parallel plates 410 are used as shearing elements to make differential displacements between the lenslet arrays. In the embodiment shown in FIG. 4 c, the optical parallel plates 410 also act as beam splitters.

The differential Shack-Hartmann curvature sensor 400 of the present invention uses three lenslet arrays 402, 404, 406 in three channels and three CCD cameras 412, 414, 416 to record the coordinates of the each Hartmann grid points as shown in FIG. 4 a. Since the lenslet 402, 404, and 406 are recorded by different CCD cameras, a reference light beam is required to calibrate the system to obtain the unknown constants in Equation (2) and Equation (4).

The calibration diagram is shown in FIG. 5. First, it is necessary to make sure that lenslets 404 and 406 are conjugate to lenslet 402 and that the shearing differential values in the x- and y-directions are S_(x) and S_(y). A reference beam is introduced by an ideal flat mirror, and the coordinates of the grid points of the three Hartmann grids are recorded for use as references to compute the constant values c_(0,x)(i), c_(0,y)(i), c_(0,yx)(i) and c_(0,xy)(i) with Equations (3) and (5).

B. Principal Curvature Computations

The normal curvature is the change of the surface normal in an osculating plane, and the principal curvatures of a non-umbilical point are the maximum and minimum values of normal curvatures, say κ₁ and κ₂, in two perpendicular directions. Regarding a local patch of surface, the principal curvatures are invariants, which are insensitive to the surface orientation. In order to evaluate the principal curvatures, it is assumed that the neighborhood of a Hartmann grid point is represented by a “Monge patch” of the form: X=x{right arrow over (e₁)}+y{right arrow over (e₂)}+W(x,y){right arrow over (e ₃)},  (6) where ({right arrow over (e₁)}, {right arrow over (e₂)}, {right arrow over (e₃)}) is an orthogonal frame in Euclidean 3-space. Then the second fundamental form has a matrix to describe the local surface shape as

$\begin{matrix} {{{II} = \begin{pmatrix} {\overset{\Cap}{\omega}}_{1}^{13} & {\overset{\Cap}{\omega}}_{1}^{23} \\ {\overset{\Cap}{\omega}}_{2}^{13} & {\overset{\Cap}{\omega}}_{2}^{23} \end{pmatrix}},} & (7) \end{matrix}$ where {circumflex over (ω)}_(j) ^(i3)(i=1,2;j=1,2.) defines the component in {right arrow over (e)}_(i) of the turning rate of the normal as the frame moves across the given point along {right arrow over (e_(j))}. For a wave front traveling in the z-direction ({right arrow over (e₃)}), W(x,y) is the “height” as a function of x and y in the pupil plane. Then at each “Monge patch”, the matrix II becomes

$\begin{matrix} {{{II} = \begin{pmatrix} {c_{x}(i)} & {c_{yx}(i)} \\ {c_{xy}(i)} & {c_{y}(i)} \end{pmatrix}},{i = 1},2,{\ldots\mspace{14mu} m},} & (8) \end{matrix}$ where the diagonal terms c_(x)(i) and c_(y)(i) are the wave front normal curvatures in the x-direction and in the y-direction, i.e.

${c_{x}(i)} = {\frac{\partial^{2}W}{\partial x^{2}}(i)}$ and

${{c_{y}(i)} = {\frac{\partial^{2}W}{\partial y^{2}}(i)}};$ off-diagonal terms c_(xy)(i) and c_(yx)(i) are the corresponding twist curvature terms, i.e.

${c_{xy}(i)} = {\frac{\partial^{2}W}{{\partial x}{\partial y}}(i)}$ and

${c_{yx}(i)} = {\frac{\partial^{2}W}{{\partial y}{\partial x}}{(i).}}$ It is assumed that c_(xy)(i)=c_(yx)(i).

The determinant of matrix II, denoted as K, is known as the Gaussian curvature. The trace of the matrix II, denoted as 2H, is known as the mean curvature. Both Gaussian curvature and mean curvature are algebraic invariants, which do not change with rotation of the orthogonal frame ({right arrow over (e₁)}, {right arrow over (e₂)}, {right arrow over (e₃)}) about the normal.

By diagonalizing the matrix II to rotate the orthogonal frame about {right arrow over (e₃)}, the off-diagonal terms disappear, and a new matrix II′ is obtained by II′=P^(T)IIP  (9) where P is an orthogonal matrix defined by

$\begin{matrix} {{P = \begin{bmatrix} {\cos\;\theta} & {{- \sin}\;\theta} \\ {\sin\;\theta} & {\cos\;\theta} \end{bmatrix}},} & (10) \end{matrix}$ where angle θ is defined as the frame rotation angle. The new matrix II′ is a diagonal matrix, which is

$\begin{matrix} {{{II}^{\prime} = \begin{pmatrix} {\kappa_{1}(i)} & 0 \\ 0 & {\kappa_{2}(i)} \end{pmatrix}},} & (11) \end{matrix}$ where κ₁(i) and κ₂(i) (κ₁(i)>κ₂(i)) are the eigenvalues of the matrix II, also known as the first and second principal curvatures that define the maximum and minimum normal curvatures at a given point patch, and i=1, 2, . . . , m, where m is the total number of Shack-Hartmann grid points. Plug the Equations (8), (10) and (11) into Equation (9), to obtain the principal curvatures κ₁ and κ₂ at each grid point as

$\begin{matrix} {{{\kappa_{1,2}(i)} = \frac{{c_{x}(i)} + {{c_{y}(i)} \pm \sqrt{\left( {{c_{x}(i)} - {c_{y}(i)}} \right)^{2} + {4{c_{xy}(i)}^{2}}}}}{2}},} & (12) \end{matrix}$ and the rotation angle θ is the angle between the first principal curvature and the x-direction, which can be obtained by

$\begin{matrix} {{\theta(i)} = {\frac{1}{2}{{\tan^{- 1}\left( \frac{2{c_{xy}(i)}}{{c_{x}(i)} - {c_{y}(i)}} \right)}.}}} & (13) \end{matrix}$ The principal curvatures can also be computed by evaluating the eigenvalues of matrix II with its characteristic equation as det(κI−II)=0,  (14) the result is the same as Equation (12). The rotation angle θ can also be computed with Euler's formula (1760) by

$\begin{matrix} {{{\cos\;{2\left\lbrack {\theta(i)} \right\rbrack}} = \frac{{2{c_{x}(i)}} - {2{H(i)}}}{{\kappa_{1}(i)} - {\kappa_{2}(i)}}},} & (15) \end{matrix}$ where H is the mean curvature given by H(i)=(c _(x)(i)+c _(y)(i))/2,  (16) Then the angle θ is given by

$\begin{matrix} {{\theta(i)} = {\frac{1}{2}{{\cos^{- 1}\left( \frac{{c_{x}(i)} - {c_{y}(i)}}{{\kappa_{1}(i)} - {\kappa_{2}(i)}} \right)}.}}} & (17) \end{matrix}$ Apply Equation (12) into Equation (17), to obtain

$\begin{matrix} {{{\theta(i)} = {\frac{1}{2}{\cos^{- 1}\left( \frac{{{c_{x}(i)} - {c_{y}(i)}}}{\sqrt{\left( {{c_{x}(i)} - {c_{y}(i)}} \right)^{2} + {4{c_{xy}(i)}^{2}}}} \right)}}},} & (18) \end{matrix}$ which is equivalent to Equation (13). C. Comparisons with the Previous Arts

With the system calibrated, the discrepancies between the image de-magnifying systems are cancelled, the discrepancies between the two arms in the cube beam splitter are cancelled, and the aberrations in the collimator and the cube prisms are also cancelled. Besides the error from the ideal flat mirror used for introducing the reference beam, which can be very small, and the nominal errors in applying the shearing differential values in Equations (2) and (3), the remaining error sources are the discrepancies between lenslets 402, 404 and 406, which are negligible for high quality micro-lenslet arrays.

As shown in the configuration in FIG. 6, a point light source 610 is used to generate the beam for measurement and the point light source 620 is used to generate the beam for calibration. After calibration, no reference light beam is necessary.

The following description compares the differential Shack-Hartmann curvature sensor with a prior art curvature sensor. In 1988, Francois Roddier proposed a method to measure the local curvature of the wavefront surface by measuring the difference in illumination of the two planes before and after the focal point as shown in FIG. 7. The Roddier' curvature sensor is based on the irradiance Poisson equation, which is derived from Teague's irradiance transport equation. Roddier obtained

$\begin{matrix} {\frac{I_{1} - I_{2}}{I_{1} + I_{2}} = {\left( {{\frac{\partial W}{\partial n}\delta_{c}} - {P{\nabla^{2}W}}} \right)\Delta\; z}} & (5) \end{matrix}$ where Δz is the distance from the pupil plane of the defocused plane P₁ or P₂ viewed from the object space. A plane at a distance from the focus is conjugated to a plane at a distance Δz from the pupil. Roddier proved that

$\begin{matrix} {{\Delta\; z} = \frac{f\left( {f - l} \right)}{l}} & (5) \end{matrix}$ So Roddier obtained the well-known equation

$\begin{matrix} {\frac{I_{1} - I_{2}}{I_{1} + I_{2}} = {\frac{f\left( {f - l} \right)}{l}\left( {{\frac{\partial W}{\partial n}\delta_{c}} - {P{\nabla^{2}W}}} \right)}} & (6) \end{matrix}$

Curvature sensing is a technique used typically in adaptive optics to measure the Laplacian of the wavefront by subtracting intensity profiles from an exact distance before and after the focus of a lens. The idea of the Roddier's curvature sensor is that the normalized differential intensity change along the optical axis provides the information of the local Laplacian curvature of the wavefront.

For adaptive optics systems, the image before and after the focus is usually switched mechanically, making the systems quite noisy during operation. In contrast, the differential Shack-Hartmann curvature sensor contains no moving parts making it more reliable.

Interferometry is a technique to make the wavefront to interfere with itself or an ideal wavefront. It is especially good for measuring high spatial frequency aberrations and low amplitude aberrations. But air motion and mechanical vibrations make obtaining an image with an interferometer difficult, especially for testing large optics. Sophisticated software is necessary to extract meaningful and accurate information from interferograms. Commercial interferometers are typically expensive, where a high quality Shack-Hartmann wavefront sensor is typically much less expensive.

Foucault knife-edge testing involves moving a knife-edge through the focus of a beam and observing the intensity pattern on a screen. Like interferometry, knife-edge testing allows high-spatial frequency aberrations to be observed. But it requires very accurate alignment of the knife-edge to the beam focus, and it is qualitative test.

In summary, the differential Shack-Hartmann Curvature sensor shares the important features of the Shack-Hartmann sensor, such as it is a real-time wavefront measurement, measurements are inherently two-dimensional and parallel, it is independent of higher-order aberrations and intensity profile variations, has good photon efficiency and is good for the all wavelength bands.

The differential Shack-Hartmann curvature sensor also provides some unique features such as eliminating the need for external references after calibration, the sensor is independent of vibrations, tilt and whole body movements, which makes it a good choice for measurements with moving objects, and is scale tunable by changing differential values.

D. Mid-Spatial Frequency Error Recovery

In another embodiment, for wave front testing with mid-spatial frequency error recovery increased spatial sampling is used. When spatial sampling with a given lenslet array is not adequate for mid-spatial frequency recovery, the sampling grid is moved with a displacement less than the lenslet pitch size to increase the sampling density. FIG. 8 shows a sampling grid from a Differential Shack Hartmann sensor at (0, 0). For error recovery, a denser sampling is accomplished by first shifting the sampling grid of FIG. 8 by 0.6 along the x-axis to a position of approximately (0.6, 0) to double the number sampling points. By shifting the sampling grid of FIG. 9 by 1.2 along the x-axis to approximately (1.2, 0) triples the number of sampling points as shown in FIG. 10. At this point, the sampling is 3 times denser in the x-direction than the sampling shown in FIG. 8. To further increase the density, the sampling grid of FIG. 10 is shifted by −0.6 along the y-axis to approximately (0, −0.6) as shown in FIG. 11, then is shifted by −1.2 along the y-axis to approximately (0,−1.2) shown in FIG. 12, The sampling grid shown in FIG. 12 is three times denser than the original sampling grid in y-direction.

In a preferred embodiment, to achieve a sampling grid that is nine times denser than the original grid. First the grid shown in FIG. 12 is again shifted by 1.2 along the x-axis to approximately 1.2, −0.6 as shown in FIG. 13. Then the grid is shifted by −0.6 along the y-axis to approximately 0, −1.2 as shown in FIG. 14. Next, the grid shown in FIG. 14 is shifted by 0.6 along the x-axis to approximately 0.6, −1.2 as shown in FIG. 15. The final shift, shown in FIG. 16, shows the grid shifted right by 0.6 along the x-axis to approximately 1.2, −1.2, which increases the density of the sampling grid by nine times.

For wave front testing with mid-spatial frequency error, increasing the sampling points to nine times denser than the original sampling grip, provides wave front mid-spatial frequency error recovery. The prior art slope- or curvature-based wave front sensors that provide measurements are insensitive to vibration can benefit from the present invention using the mid-spatial frequency error recovery. The systems and methods of the present invention can be applied for different wave front sensing using a variety of sampling-grid-sensors including but not limited to Shack-Hartman sensors, curvature sensors, and shearing interferometers. Alternative sensors and techniques, such as 4-D technique will be obvious to those skilled in the art.

While the invention has been described, disclosed, illustrated and shown in various terms of certain embodiments or modifications which it has presumed in practice, the scope of the invention is not intended to be, nor should it be deemed to be, limited thereby and such other modifications or embodiments as may be suggested by the teachings herein are particularly reserved especially as they fall within the breadth and scope of the claims here appended. 

1. A differential sensing device for measuring a wavefront curvature using increased spatial sampling for wavefront testing comprising: a sampling sensor having an output beam; an optical element to split said output beam into a first, second and third beam, each of said first, second and third beams traveling in at least one of an x, y and z direction, respectively; a first, second and third lenslet array in the path of the first, second and third beam, respectively, to generate a corresponding first, second and third sampling grid each having plural grid points; a shearing element for shifting said plural grid points a distance in a horizontal and a vertical direction to produce plural sampling grids each having plural grid points the displacement of the shifting being less than a pitch size of the lenslet array; and a measuring device for measuring plural slopes of plural wave fronts at each grid point on the spatial sampling grid to obtain wave front normal curvatures and corresponding twist curvature terms for determining a principal curvature and direction, wherein said plural slopes at each of the plural grid points is measured approximately simultaneously before and after differentially shifting the sampling grid.
 2. The device of claim 1, wherein each of said first and second shearing elements comprise: an opto-mechanical device.
 3. The device of claim 1, wherein each of said first and second shearing elements comprise: an electro-optics device.
 4. The device of claim 1, wherein each of said first and second shearing elements comprise: an optical parallel plate to make the differential displacements between the first and the second lenslet array.
 5. The device of claim 1, wherein the first and second shearing elements comprise: a first and a second micro-screw.
 6. The device of claim 1 said measuring device comprises: a first, second and third camera in a corresponding path of each of said first, second and third beams, respectively, for measuring said wave front slopes at each one of the plural sampling grid point.
 7. The device of claim 1, wherein the sampling grid sensor comprises: a Shack-Hartman slope sensor.
 8. The device of claim 1, wherein the sampling grid sensor comprises: a shearing interferometer sensor.
 9. A system for measuring differentials of wave front slopes to determine a wavefront curvature without the use of a reference, the system comprising: a light source for generating an input light beam; a beam splitter splitting the input light beam to produce at least three output beams in different directions; at least three lenslet arrays, one in a path of each of the at least three output beams for generating a corresponding at least three sampling grids; at least one optical element for shearing two of the at least three output beams in a horizontal direction and a vertical direction for shifting said two of the at least three sampling grids plural differential distances in said horizontal direction and vertical directions; a measuring system for measuring a slope at each one of plural grid points generated by said at least three output beams; and a processor for executing a set of instructions for calculating a normal curvature and a principal curvature of a wavefront from said slopes measured at each of the plural grid points for wavefront testing.
 10. The system of claim 9, wherein at least one optical element comprises: a first and a second optical parallel plates for making differential displacements between the at least three and the at least three lenslet arrays.
 11. The system of claim 9, wherein the beam splitter comprising: a first and second optical parallel plate to act as the beam splitter for splitting the input beam into the first, second and third output beams and act as the at least one optical element to make differential displacements between the first, second and third lenslet arrays, wherein the first and third lenslet arrays shear each other in the first direction and the second and third lenslet arrays shear each other in the second direction.
 12. The system of claim 9, wherein said measuring system comprises: a first, second and third camera in each of said first, second and third beam, respectively, for recording approximately simultaneously said plural coordinates of said plural grid points at each one of the plural grid points of the first, second and third grids to obtain slope differentials.
 13. The system of claim 9, wherein said set of instructions comprises: a first subset of instructions to obtain a derivative of the plural wave front slopes at the plural grid points to compute said wavefront local curvatures and principal curvatures and corresponding directions.
 14. A method for measuring a normal curvature of a wavefront with mid-spatial frequency error detection with a differential curvature sensor comprising the steps of: providing a sensing system having a first, second and third output beam in different directions for generating an original sampling grids having plural grid points; measuring differentials of the plural wave front slopes at plural grid points of the original sampling grid; moving the sampling grid in a horizontal direction with a displacement less than a pitch size of the lenslet compared to said original sampling grid; measuring differentials of the plural wave front slopes at plural grid points of the horizontally displaced grid to produce a second sampling grid; moving the original grid in a vertical direction with a displacement less than a pitch size of the lenslet compared to said original sampling grid and the second sampling grid to produce a third sampling grid; measuring differentials of the plural wave front slopes at plural grid points of the third sampling grid; repeating the horizontal and vertical moving and measuring steps four or more times to a different coordinate for increased spatial sampling; and determining said normal curvature and twist curvature terms of said wavefront from said measured differentials of said plural wave front slopes for wavefront testing with mid-frequency error recovery.
 15. The method of claim 14 wherein the normal curvature determination step comprises the step of: applying ${c_{x}(i)} = {{\frac{1}{f}\left( \frac{x_{i}^{\prime\;{med}} - x_{i}^{med}}{s_{x}} \right)} - {c_{o,x}(i)}}$ and ${{c_{y}(i)} = {{\frac{1}{f}\left( \frac{{y^{''}}_{i}^{mea} - y_{i}^{mea}}{s_{y}} \right)} - {c_{0,y}(i)}}},$ where c_(0,x)(i) and c_(0,y)(i) are obtained by ${c_{o,x}(i)} = {\frac{1}{f}\left( \frac{x_{i}^{\prime\;{Ref}} - x_{i}^{Ref}}{s_{x}} \right)}$ and ${{c_{o,y}(i)} = {\frac{1}{f}\left( \frac{y_{i}^{''\;{Ref}} - y_{i}^{Ref}}{s_{y}} \right)}},$ to the measured coordinate differentials to obtain said normal curvature of said wavefront.
 16. The method of claim 14, wherein the twist curvatures determination step comprises the step of: applying ${{c_{xy}(i)}\frac{1}{f}\left( \frac{x_{i}^{''\;{mea}} - x_{i}^{mea}}{s_{y}} \right)} - {c_{0,{xy}}(i)}$ and ${{c_{yx}(i)} = {{\frac{1}{f}\left( \frac{y_{i}^{\prime\;{mea}} - y_{i}^{mea}}{s_{x}} \right)} - {c_{0,{yx}}(i)}}},$ where c_(0,yx)(i) and c_(0,xy)(i) are constants given by ${c_{0,{yx}}(i)} = {\frac{1}{f}\left( \frac{y_{i}^{\prime\;{Ref}} - y_{i}^{Ref}}{s_{x}} \right)}$ and ${{c_{0,{xy}}(i)} = {\frac{1}{f}\left( \frac{x_{i}^{''\;{Ref}} - x_{i}^{Ref}}{s_{y}} \right)}},$ to the measured coordinate differentials to obtain said twist curvatures of said wavefront.
 17. The method of claim 14 further comprising the step of: calibrating said sensing system to reduce systematic errors.
 18. The method of claim 14 wherein said differential measurement step comprises the steps of: measuring coordinates of plural grid points of the plural wavefront without said differential displacement; and measuring coordinates of plural grid points of the plural wavefronts with said differential displacement.
 19. The method of claim 14, wherein the repeating the horizontal and vertical moving steps comprises the steps of: measuring differentials of the original grid points of the original sampling grid; moving the original sampling grid horizontally a first positive horizontal distance to increase a density of the sampling grid points in a horizontal direction; measuring differentials of the first horizontally shifted grid points; moving the first horizontally shifted sampling grid horizontally a second positive horizontal distance to further increase the density of sampling grid points in the horizontal direction; measuring differentials of the second horizontally shifted grid points; moving the original sampling grid vertically a first negative vertical distance to increase the density of the sampling grid points in vertical direction; measuring differentials of the first vertically shifted grid points; moving the first vertically shifted sampling grid vertically a second negative vertical distance to further increase the density of sampling grid points in vertical direction; and measuring differentials of the second vertically shifted grid points.
 20. The method of claim 19, further comprising the steps of: moving the first horizontally shifted sampling grid vertically a first negative vertical distance to increase the density of the sampling grid points; measuring differentials of the first horizontally and vertically shifted grid points; moving the first horizontally and vertically shifted sampling grid a second horizontal positive direction to increase the density of the sampling grid points; measuring differentials of the second horizontally and vertically shifted grid points; moving the first horizontally and vertically shifted sampling grid a second negative vertical direction to increase the density of the sampling grid points; measuring differentials of the first horizontal and second vertically shifted grid points; moving the second horizontal and vertically shifted sampling grid a second negative vertical direction to increase the density of the sampling grid points; and measuring differentials of the second horizontally and second vertically shifted grid points. 